Crib sheet: properties of morphisms of schemes originally by Johan de Jong; adapted by Kiran Kedlaya

Surgeon General’s Warning: Not reading EGA before using this crib sheet may be hazardous to your health. In particular, certain definitions (e.g., projective) do not match Hartshorne.

Here are some properties of a morphism f : X → Y of schemes. All of these are Zariski local on the base unless otherwise specified.

open immersion On topological spaces, f is a homeomorphism of X with an open subset of Y. [ ⟹ étale]

closed immersion On topological spaces, f is a homeomorphism of X with a closed subset of Y, and the map f^{♯} : O_{Y} → f_{∗}O_{X} is surjective. [ ⟹ finite]

quasi-compact For every affine open V ⊆ Y, f^{−1}(V) is a quasi-compact topological space. Equivalent: for every quasi-compact open V ⊆ Y, f^{−1}(V) is a quasi-compact topological space. Equivalent: for every affine open V ⊆ Y, f^{−1}(V) is a finite union of affine opens of X.

quasi-separated The diagonal Δ_{f} : X → X ×_{Y} X is quasi-compact.

separated The diagonal Δ_{f} : X → X ×_{Y} X is a closed immersion. [ ⟹ quasi-separated]

affine For every affine open V ⊆ Y, f^{−1}(V) affine. Equivalent: X ≅ Spec_{Y} (A) for some quasi-coherent O_{Y} -algebra A. [ ⟹ separated, quasi-compact, quasi-affine]

quasi-affine f is quasi-compact and admits a factorization [figure: X; j; f; X′; f′; Y] with j an

open immersion and f′ affine. Equivalent: for every affine open V ⊆ Y, f^{−1}(V) is isomorphic to a quasi-compact open subscheme of an affine scheme.

locally of finite type/presentation For every affine open V ⊆ Y, f^{−1}(V) can be covered by affine opens U such that Γ(X, U) is a finitely generated/presented Γ(Y, V)-algebra. [local on source]

of finite type/presentation f is quasi-compact and locally of finite type/presentation.

quasi-projective f is of finite type and there exists an invertible O_{X}-module L such that L is f-ample. Consequence: for every affine open V ⊆ Y, a suitably high power of L has sections over f^{−1}(V) which define an immersion of f^{−1}(V) into ℙ_{V}^{N}. [not local on base, ⟹ separated]

proper f is separated, of finite type and universally closed. projective X is isomorphic to Proj_{Y} S, where S is a quasi-coherent graded O_{Y} -

module such that S_{0} = O_{Y}, generated in degree 1 (i.e., Sym^{∗} S_{1} → S is surjective). Equivalent: X → Y factors through a closed immersion X → Proj_{Y} (Sym^{∗} F) with F coherent over Y. [ ⟹ proper, quasi-projective]

quasi-finite f is of finite type, and for all y ∈ Y, the scheme X_{y} = f^{−1}(Y) is finite over Spec κ(y).

finite f is affine and for every affine open V ⊆ Y, Γ(O_{X}, f^{−1}(V)) is a finite Γ(O_{Y} , V)-module. Equivalent: f is proper and quasi-finite. Equivalent: f is proper and affine.

formally unramified For all diagrams (1), there exists at most one ψ̃. Equivalent: Ω_{X/Y}^{1} = 0.

formally smooth For all diagrams (1), there exists at least one ψ̃.

formally étale For all diagrams (1), there exists exactly one ψ̃.

unramified f is formally unramified and locally of finite presentation.

smooth f is formally smooth and locally of finite presentation.

étale f is formally étale and locally of finite presentation. Equivalent: f is flat and unramified.

In the following diagram, I is an ideal of the ring A such that I^{2} = 0.

\xymatrix{ &  & X \ar[dd]^{f} \\ Spec A \ar[drr] \ar@{-->}[urr]^{ψ̃} & Spec A/I \ar[ur]^{ψ} \ar@{^{(}->}[l] \ar[dr] \\  &  & Y}  (1)

Some trivial and nontrivial results:

1. A scheme X is (separated/affine) iff X → Spec ℤ is.

2. f is smooth iff f is locally of finite presentation, flat and has smooth fibres.

3. If f is of finite type and formally unramified, then f is quasi-finite.

4. For any scheme X, the diagonal Δ : X → X ×_{Spec ℤ}X is separated and locally of finite type. If X is quasi-separated, then Δ is of finite type and also quasi-affine.

5. If f is quasi-finite and separated, then f is quasi-affine.

6. (Zariski’s Main Theorem) If f is quasi-finite and separated and Y is quasi-compact and quasi-separated, then there exists a factorization f = f′ ∘ u with u : X → X′ an open immersion and f′ : X′ → Y finite.

7. (Nagata’s Theorem) If f is of finite type and separated and Y is Noetherian, then there exists a factorization f = f′ ∘ u with u : X → X′ an open immersion and f′ : X′ → Y proper. (This presumably also holds with Y quasi-compact and quasi-separated.)